The parabola and its directrix

Definition: A parabola is the set of points that are equidistant from a point, called the focus point, and a line, called the directrix or the directrix line.

To do and explore

First we can see how this image was formed: Double click the image, and use the bottom menu to follow the construction. I started with the point F and then the directrix line. The directrix point sits on the line. A line segment is drawn from F to the directrix point, and a perpendicular bisector, L1, to this segment. L2 is constructed through the directrix point perpendicular to the directrix line. The point T is the intersection between L1 and L2. Now the locus of T as I move the directrix point is a parabola, which we will see. L1 is always a tangent line to the parabola.

Since L1 is a perpendicular bisector of FP, the triangle FTP has two legs equal. According to our definition, the locus is a parabola.

Note, that the opposite angles between L1 and L2 are equal, and therefor the angle between the line segment FT and L1 also equals the angle between the lines. If L2 is an incomming ray that is reflected in the parabola, it will be reflected towards F.